Boundedness of solutions to a quasilinear parabolic-parabolic Keller-Segel system with a logistic source

We study global solutions of a class of chemotaxis systems generalizing the prototype {u(t) = del.(phi(u)del u) - chi del.(Psi(u)del v) + au - bu(r), x is an element of Omega, t > 0, v(t) = Delta v - v + u, x is an element of Omega, t > 0 in a bounded domain Omega subset of R-N (N >= 1) with smooth boundary partial derivative Omega, phi(u) = (u+1)(-alpha), Psi(u) = u(u+1)(beta-1), and the parameters r > 1, a >= 0, b, chi > 0, and alpha, beta is an element of R. It is proved that if 0 < alpha+beta < max{r - 1 + alpha, 2/N}, or b is big enough, if beta = r - 1, then the classical solutions to the above system are uniformly-in-time bounded. Our results improve the results of Wang et al. (2014) [28] and Cao (2014) [3] and also enlarge the range of the results of Tao and Winkler (2012) [25] and Ishida et al. (2014) [14]. (C) 2015 Elsevier Inc. All rights reserved.